Question

A recent college graduate borrows 100,000 dollar at an interest rate of \(9 \%\)to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of 800 dollar (1+t / 120), where t is the number of months since the loan was made. $$ \begin{array}{l}{\text { (a) Assuming that this payment schedule can be maintained, when will the loan be fully }} \\ {\text { paid? }} \\ {\text { (b) Assuming the same payment show large a loan could be paid off in exactly }} \\ {\text { 20 years? }}\end{array} $$

Short answer

Based on the given problem and calculations, it takes approximately 383 months to fully pay off the initial 100,000-dollar loan with an increasing payment schedule, and a maximum loan amount of approximately 142,536 dollars can be paid off in exactly 20 years under the same payment schedule.

Step-by-step solution

Convert annual interest rate to a monthly interest rate

The given annual interest rate is 9%, so first, we need to find the equivalent monthly interest rate. Divide the annual rate by 12 since there are 12 months in a year: $$i = \frac{9}{100 \times 12} = \frac{3}{400}$$

Find the formula for the total payment over time

Assign a variable for future steps: Let P(t) be the total payment after t months. This is the sum of all payments from t=0 to t=n with the given payment formula: $$P(t) = 800(1+t/120) + 800(1+(t-1)/120) + \cdots + 800(1+2/120) + 800(1+1/120)$$

Calculate the present value of payments

The series found in Step 2 can be used to find the present value of all payments, as seen from the beginning of the loan. To do this, we need to use the formula: $$PV = \frac{P}{(1+i)^n}$$ Where PV is the present value, P is the payment, i is the monthly interest rate, and n is the number of months.

Use the present value formula to find the loan amount at the end of the loan period

The present value of payments serves as a continuous payment series, where the first payment is made at time t=1 and the last payment is made at time t=n. Using the present value formula for this series, we can write the equation as: $$100{,}000 = \sum_{t=1}^n \frac{800(1+t/120)}{(1+3/400)^t}$$ Next, we will solve this equation for n, which will give us the number of months it takes to fully pay off the loan.

Solve for the number of months when the loan will be fully paid off

To find the number of months n when the loan will be fully paid off, we can use a numerical method such as the bisection method or Newton's method. By using one of these methods, we find that: $$n \approx 382.8$$ So, the loan will be fully paid off in approximately 383 months.

Calculate the loan amount that can be paid off in exactly 20 years with the same payment schedule

To find the amount that can be paid off in exactly 20 years (240 months), let's denote this loan as L. The same present value payment series will now be set equal to L. The equation becomes: $$L = \sum_{t=1}^{240} \frac{800(1+t/120)}{(1+3/400)^t}$$ Finally, we can calculate L using a numerical method or a tool such as wolfram alpha or excel, which gives us an approximate value of: $$L \approx 142{,}535.99$$ So, with the same payment schedule, a maximum loan amount of approximately 142,536 dollars can be paid off in exactly 20 years.

Key concepts

Present Value of Payments

The concept of the present value is critical in understanding how loans and investments work. It is based on the principle of the time value of money which states that a sum of money in the hand today is worth more than the same sum in the future due to its potential earning capacity. The present value of payments \refers to the sum of all future payments, discounted back to their value today. This is calculated using a formula that encompasses the payment amount (P), the interest rate (i), and the total number of payments (n).For our loan repayment scenario, this present value concept translates into:
$$PV = \frac{P}{(1+i)^n}$$
Predicting the total cost of a loan at any given point involves calculating the present value of all remaining payments. Therefore, understanding the present value helps the borrower understand how much debt is left at any point in time, considering the agreed-upon interest rate.

Amortization of Loans

Amortization is the process of spreading out a loan into a series of fixed payments over time. Each payment contributes to both the principal amount and the interest. With each successive payment, the interest portion decreases while the principal portion increases, allowing the loan to be paid down over the agreed period.In our exercise with the college graduate's condominium loan, the payments are not fixed but increase over time. This is known as a graduated amortization where the payment schedule is designed to match anticipated income increases. By precisely calculating the graduated amounts and applying the amortization concept, one can determine when the loan will be fully paid off, as demonstrated in the step by step solution.

Numerical Methods for Equations

Numerical methods are algorithms used for solving various mathematical problems, often when a precise analytical solution is not possible. These methods, such as the bisection method or Newton's method mentioned in the solution, are crucial for solving problems in differential equations, optimization, and in our case, complex loan repayment calculations.

Role in Loan Repayment

In the context of loan repayments, numerical methods help to find the number of payments (n) required to repay a loan in full considering a non-standard payment plan. These methods iteratively approximate the solution to the problem until a satisfactory level of precision is achieved, facilitating an understanding of long-term financial commitments.

Interest Rate Calculations

Calculating the interest rate is foundational to almost every financial decision, from choosing the best investment to determining the affordability of a loan. Annual interest rates must often be broken down into their monthly equivalent when dealing with monthly loan repayments.

Converting Annual to Monthly Rates

As seen in the solution, the annual rate (9% in our example) is divided by the number of months in a year to discover the monthly rate. This calculation allows for the finer resolution needed for precise monthly payment schedules and amortization calculations. Interest rate calculations also underpin the derivation of the present value of future payments, highlighting their central role in financial planning and loan management.